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arXiv · 1909.08324

On infinitesimal generators of sublinear Markov semigroups

Franziska Kühn
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Abstract

We establish a Dynkin formula and a Courr\`ege-von Waldenfels theorem for sublinear Markov semigroups. In particular, we show that any sublinear operator $A$ on $C_c^{\infty}(\mathbb{R}^d)$ satisfying the positive maximum principle can be represented as supremum of a family of pseudo-differential operators: $$Af(x) = \sup_{\alpha \in I} (-q_{\alpha}(x,D) f)(x).$$ As an immediate consequence, we obtain a representation formula for infinitesimal generators of sublinear Markov semigroups with a sufficiently rich domain. We give applications in the theory of non-linear Hamilton--Jacobi--Bellman equations and L\'evy processes for sublinear expectations.

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Franziska Kühn. 2019-09-18. On infinitesimal generators of sublinear Markov semigroups. https://arxiv.org/abs/1909.08324

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